Guessing what a simple partial differential equation is describing physically

Question

Is there an easy way to look at a partial different equation and get a sense of what kind of phenomena it is physically describing? I have an equation that looks like this:

$$\partial_tA=C_3\partial^2_{xx}A+C_2\partial_x\left(vA\right)+C_1A+C_0.$$

From inspecting it and doing some searching, it seemed like it might be trying to describe a phenomena that had advection and diffusion occurring at the same time. Is this a correct interpretation?

Note that $C_3>0$, $C_2<0$, $C_1<0$, $C_0>0$ and $v(x,t)$ is velocity. I'm not sure, but it seems reasonable to assume velocity can be either negative or positive. Also, $A(x,t)$ should be positive.

Note that I'm not looking for a solution to the equation, just the physical interpretation of what kind of physics would be involved.

Answer 1

Let us try to rewrite the equation in approximate form of finite differences: $$\frac{A(x,t+\Delta t)-A(x,t)}{\Delta t} = C_3\frac{A(x+h,t)+A(x-h,t)-2A(x,t)}{h^2} +$$ $$+ C_2 \frac{v(x+h,t)A(x+h,t)-v(x-h,t)A(x-h,t)}{2h} + C_1 A(x,t)+C_0$$ Where $\Delta t$ -- is a time step, and $h$ -- space step.
The expression becomes your PDE, in the limit $\Delta t\to0, h\to0$.

The left hand side describes how much the quantity $A$ changes during a time step at a given point (hope that is obvious). Let us see what is on right hand side term-by-term:

• $C_3$: If $A$ is larger in total at "neighboring" points than in the point itself: $A(x+h,t)+A(x-h,t)>2A(x,t)$ then $A$ will increase at $x$. Otherwise it will decrease. So the term "forces" $A$ to some kind of local equilibrium.
  This is a standard term for description of various diffusion or heat distribution processes that forces your system to some equilibrium.
• $C_2$: Here one have to be careful with signs. Let us call $x+h$ "the point to the right" and $x-h$ will be "the point to the left". If $v>0$ then it "flows to the right" and if $v<0$ it "flows to the left". Now you check that if your flow goes into your point $x$ (say, flows to the right from point to the left) then it increases $A$ at $x$. And decreases if flows goes away from your point.
  That is a standard term that describes stuff carried with the flow $v$ and usually derived in terms of substantial derivative
• $C_1$ and $C_0$: These two are trivial, because they are local. $C_0$ just gives a constant contribution to growth of $A$, and $C_1$ changes (inhibits in your case) the growth rate proportional to the value of $A$.
  You can also understand them in terms of local ODE for a given point: $\dot{a} = C_1a+C_0$

To sum it up: let us say that $A(x,t)$ describes density of, say, bacteria in a tube. Then $C_3$ describes how they diffuse around, $C_2$ describes how they are carried by a stream $v(x,t)$ of liquid in the tube, $C_0$ -- is a growth rate of new bacteria, and, finally, $C_1$ -- is responsible for the growth to slow down from the overpopulation.

Answer 2

As an alternative to Christian Blatter's heat interpretation, $A$ might describe the concentration of particles adsorbed onto a one-dimensional substrate surface (or a two-dimensional one, where we ignore one of the dimensions).

• New particles are adsorbed at rate $C_0$ per unit length.
• Adsorbed particles detach from the surface at rate $-C_1$ per particle.
• The particles move along the surface (or the surface itself moves) with mean net velocity $-C_2 v$.
• While moving, the particles also diffuse over the surface with diffusion coefficient $C_3$.

In any case, this equation describes the dynamics of some quantity $A$ undergoing diffusion and advection over a one-dimensional space, while also undergoing constant (i.e. zeroth-order) local accumulation and first-order decay.

(As a mathematical ecologist, my first thought was to interpret it as a spatial population model, but it doesn't really fit that interpretation very well: there's no $A^2$ term that could describe local density regulation.)

Answer 3

You have a thin cylindrical tube along the $x$-axis that is filled with some gas of density $\rho(x,t)$. The temperature of the gas is $A(x,t)$, and the gas is moving along with "mass flux" $m(x,t):=\rho(x,t)v(x,t)$, where $v$ denotes the actual speed of individual particles. (The $\rho$ is missing in your equation). Heat is transported through heat conduction and by means of convection. In addition the $x$-axis is an electrical wire which produces heat hat a constant rate, and at the surface of the tube we have a heat loss towards outer space, the latter being a temperature $0$.

Your equation describes the temporal change rate of temperature in a "length element" at $x$ at time $t$. The individual terms on the right side account for the contributions of conduction, convection, surface loss to outer space, and heating.

Answer 4

Neglecting terms and solving the equation for idealized initial conditions is one way to study what each term means.

$$∂_tA=C_3∂_x^2A+C_2∂_x(vA)+C_1 A+C_0$$

For example, set all but $C_0$ to zero and obtain $A = C_0 t + A(x,0)$. The term with $C_0$ supplies the quantity represented by $A$ at a constant rate. It's a source term.

Set all but $C_1$ to zero and obtain $A = A(x,0) e^{C_1 t}$. This is a kind of nonlinear source term -- $A$ is supplied at a rate proportional to its quantity in a positive feedback.

Set all but $C_2$ to zero and you find $A = A(x,0)\delta(x-C_2 vt)$ assuming $v$ is constant. You can generalize for $v$ x-dependent. This is an advection term which the describes the movement of $A$ at velocity $C_2 v$.

Set all but $C_3$ to zero and you find the classic diffusion equation. Here the solution for a point source initial condition is something like $A(x,t) = \frac{1}{\sqrt{C_3 t}} e^{-x^2/(C_3 t)}$ although I have some constants wrong. This describes a "spreading out" or smearing tendency due to differential transport characteristics between components of $A$.

Long story short, you have diffusion with diffusivity $C_3$, advection with velocity $C_2 v$, nonlinear supply at rate $C_1$, and constant supply at rate $C_0$.